This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.
I haven't read the article, but my understanding is that a normal curve results from summing several samples from most common probability distributions, and also a normal curve results from summing many normal curves.
All summation roads lead to normal curves. (There might be an exception for weird probability distributions that do not have a mean; I was surprised when I learned these exist.)
Life is full of sums. Height? That's a sum of genetics and nutrition, and both of those can be broken down into other sums. How long the treads last on a tire? That's a sum of all the times the tire has been driven, and all of those times driving are just sums of every turn and acceleration.
I'm not a data scientist. I'm just a programmer that works with piles of poorly designed business logic.
How did I do in my interview? (I am looking for a job.)
Say I have N independent and identically distributed random variables with finite mean. Assuming the sum converges to a distribution, what is the distribution they converge to?
If I had made the extra condition that the random variables had finite variance, you'd be correct. Without the finite variance condition, the distribution is Levy stable.
Levy stable distributions can have finite mean but infinite variance. They can also have infinite mean and infinite variance. Only in the finite mean and finite variance case does it imply a Gaussian.
Levy stable distributions are also called "fat-tailed", "heavy-tailed" or "power law" distributions. In some sense, Levy stable distributions are more normal than the normal distribution. It might be tempting to dismiss the infinite variance condition but, practically, this just means you get larger and larger numbers as you draw from the distribution.
This was one of Mandelbrot's main positions, that power laws were much more common than previously thought and should be adopted much more readily.
As an aside, if you do ever get asked this in an interview, don't expect to get the job if you answer correctly.
But if you haven't had exposure to this either through work experience or through course work it would be unfair to ask this question and use your answer to judge competence.
For a potential coworker role I would certainly be curious about your curiosity but a sharp ended question is not a way to explore that.
It's amazing that you find so many that are uncomfortable with this question. I literally teach a first-year data science course and I ask the students this very question. I spend half a lecture on it and put it in their assessment.
This is one of the most fundamental things to understand in statistics. If you don't have at least some degree of comfort with this, you have no business working with data in a professional capacity.
You can be comfortable about the concept, but not comfortable about the interview.
The way I understand it, OP asked this as a way to open the conversation, while candidates interpreted it as a math problem to solve, unintentionally getting their mind into "exam" mode.
A lot of times I can't tell if I'm the idiot or if everyone else is. Says that this isn't an interesting question at all and the article was horrible. I studied data science for a few years but I'm no expert, but it seems pretty obvious to me that if you make a series of 50/50 choices randomly, that's the shape you end up with and there's really nothing more interesting about it than that.
I don't think "obvious" is the right word here. It makes perfect sense when you understand it, but it's not a conclusion that most people could come to immediately without detailed, assisted study.
